This relation allows us to incorporate strain incompatibility arising from bulk and interfacial defect fields as well as those supported at \(O\) . We make further regularity assumptions on the defect fields.
We consider \(\Theta (\psi )=\int _\Omega \theta _B \psi \operatorname{da}+\int _S \theta _S \psi \operatorname{dl}\) , for all \(\psi \in \mathcal {D}(\Omega -O)\) , where \( \theta _B\)  and \(\theta _S\)  are bulk and interfacial densities of disclinations, respectively; \(A(\psi )=\int _\Omega \langle \alpha _B, \psi \rangle \operatorname{da}+\int _S \langle \alpha _S, \psi \rangle \operatorname{dl}\) , for all \(\psi  \in \mathcal {D}(\Omega -O,\mathbb {R}^2)\) , where \( \alpha _B\)  and \(\alpha _S\)  are bulk and interfacial densities of dislocations, respectively; and \(Q(\psi )=\int _\Omega \langle q,\psi  \rangle \operatorname{da}\) , for all \(\psi \in \mathcal {D}(\Omega -O,\operatorname{Lin})\) , where \(q\)  is the bulk density of metric anomalies. On substituting these into (REF ), and using [1]} and Identity REF , we obtain incompatibility field of the form \(N_1(\psi )=\int _\Omega {\eta } \psi \operatorname{da}+\int _S {\zeta }_{1} \psi \operatorname{dl}+\int _S {\zeta }_{2} \partial \psi /\partial \nu  \operatorname{dl}\) , for all \(\psi \in \mathcal {D}(\Omega -O)\) , such that
\(\operatorname{curl}\alpha _B + \theta _B + \operatorname{curl}\operatorname{curl}q= {\eta }~ \text{in}~ \Omega -\lbrace S\cup \lbrace O\rbrace  \rbrace , \\\frac{d}{d s}\langle \llbracket \alpha _B \rrbracket ,t \rangle + \langle \alpha _{S},\nu  \rangle + \theta _S + \left\langle \left.\nabla q\right., r \right\rangle + k \left\langle \left.q\right., \nu \otimes \nu  \right\rangle = {\zeta _1} ~\text{on}~ S-\lbrace O\rbrace , \\\langle \alpha _{S},t \rangle + \left\langle \left.q\right., t\otimes t \right\rangle = {\zeta _2}~ \text{on}~ S-\lbrace O\rbrace , and \\\Theta (v^\alpha )+A(e_3 \times \nabla v^\alpha )+Q(\mathbb {A}\nabla \nabla v^\alpha )=N_1(v^\alpha )~\text{at}~O, \) 
